Discretization of PDEs with variable coefficients using locally adaptive sparse grids

Abstract

AbstractElliptic partial differential equations with variable coefficients can be discretized on sparse grids. With prewavelets being L2‐orthogonal, one can apply the Ritz‐Galerkin discretization to obtain a linear equation system with unknowns. However, for several applications like partial differential equations with corner singularities or the high‐dimensional Schrödinger equation, locally adaptive grids are needed to obtain optimal convergence. Therefore, we introduce a new kind of locally adaptive sparse grid and a corresponding algorithm that allows solving the resulting finite element discretization equation with optimal complexity. These grids are constructed by local tensor product grids to generate adaptivity but still maintain a local unidirectional approach. First simulation results of a two‐dimensional Helmholtz problem are presented.